A conservative high‐order discontinuous Galerkin method for the shallow water equations with arbitrary topography

Abstract

SUMMARYA conservative high-order Godunov-type scheme is presented for solving the balance laws of the 1D shallow water equations (SWE). The scheme adopts a finite element Runge-Kutta (RK) discontinuous Galerkin (DG) framework. Based on an overall third-order accurate formulation, the model is referred to as RKDG3. Treatment of topographic source term is built in the DG approximation. Simplified formulae for initializing bed data at a discrete level are derived by assuming a local linear bed function to ease practical flow simulations. Owing to the adverse effects caused by using an uncontrolled global limiting process in an RKDG3 scheme (RKDG3-GL), a new conservative RKDG3 scheme with user-parameter-free local limiting method (RKDG3-LL) is designed to gain better accuracy and conservativeness. The advantages of the new RKDG3-LL model are demonstrated by applying to several steady and transient benchmark flow tests with irregular (either differentiable or non-differentiable) topography.

Topography discretization techniques for Godunov-type shallow water numerical models: a comparative study

“…The local RKDG2 method taken with the P 1 -projection of the topography verifies exactly the C-property (Kesserwani and Liang 2011). …”

“…More recently, with the establishment of the discontinuous Galerkin (DG), a local secondorder (or higher-order) accurate Godunov-type formulation may be intrinsically derived from the conservation laws of the SWE, providing a more sophisticated formulation than the traditional FV framework (Kesserwani and Liang 2011).…”

Multiwavelet-based grid adaptation with discontinuous Galerkin schemes for shallow water equations

“…From the two-scale relations for the basis functions (18) and (19) we can derive similar relations for the coefficients: the multi-scale transformation can be implemented very efficiently by recursively applying:…”

“…By this we make sure that the shape of b is involved in the adaptation process, i.e., wherever b is discontinuous or has a strong gradient the corresponding adaptive grid is refined in this area. Several other strategies to ensure well-balancing in the reference scheme rely on the continuity of b h [18]. In this case one has only to make sure that the initial projection of b to its static adaptive grid ensures the continuity.…”

“…In particular, this requires more data transfer on distributed memory architectures and, thus, affects the efficiency of parallel implementations. Discontinuous Galerkin (DG) schemes have been recently applied to the shallow water equations [16,17,18,19,20,21]. The DG method generalizes the concept of the FV method, while relying on the Finite Element notion of projecting the solution onto a space of trial functions, but, without the restriction of keeping the functions continuous.…”